Dilatational strain due to dislocations in copper

LETTERS

TO THE

EDITOR

611

density was highest in LiF crystals through which the

debris

cleavage crack progressed most slowly.

surfaces, it has not been possible to measure their rate

that the dislocations play an important by

introduced

This suggests

during cleavage may

role in the ductility.

ductile

after immersion

in hot 20%

H&30,

followed by a water rinse. Sodium chloride is ductile after surface dissolution with water, after a dip into a dry ether solution

of stearic

acid,

of surface cracking

irregularities

on freshly

by the silver deposition

Also, reliable data on rate of ductility

It was also found that the ductility can be restored a variety of surface treatments. Brittle MgO

becomes

and surface

and after the

vacuum deposition of a layer of sodium chloride or gold. Some of these treatments are of such nature

cleaved method.

change on water

polished crystals are not yet available. Thus, direct correlations are not permitted. Since it is known that a small tensile stress accelerated

the rate of cracking

on water polished crystals, it is possible that a sufficient number of cracks may form rapidly under the influence

of stress induced

by

cleavage

and cause

freshly cleaved crystals rapidly to lose their ductility.

as to suggest that it is not always necessary to remove

X-ray surface damage detectable by electron diffraction also has been found(g) to occur more rapidly on

the surface layers to restore ductility. of restoration varies with the treatment;

tion of or reaction with atmospheric

sulfuric

acid treated

The extent for example,

MgO is not as ductile

freshly cleaved crystal.

as the

On the other hand, the stearic

acid solution renders NaCl more ductile than does the water treatment, which in itself results in greater ductility

than fresh cleavage.

Another

important

but as yet unexplained

result

is that the elastic modulus is also changed by surface treatment. If the ductility is improved, the modulus is decreased. For extent of reduction

cleavage faces than on water treated faces.

Adsorp-

gases could also

result in an increase in the surface strain and thus accelerate

cracking.

Finally,

it is also possible

cracks in a range of sizes below that detectable technique

that

by our

are formed rapidly after cleavage or other

surface treatment. It is obvious that further careful experimentation is

needed

ductility

before

the

basic

in these materials

processes

controlling

can be clearly

sulfuric acid treated MgO, the varies with the time of exposme

defined.

R. A. LAD C. A. STEARNS

to the acid. Reductions in modulus as much as 50 per cent of the original value have been obtained. The

M.

G. DEL DUCA

stress-strain curves reported for NaCl by Aerts and Dekeyser”’ also appear to show a variation in the

National Advisory Committee for Aeronautics

elastic region with various specimen treatments. experiments have been reported(7t8) Recently,

Cleveland, Ohio

which give evidence that the atmosphere

1. A. A. GRIFFITH,Phil Trans. Roy. SW. A221, 163 (1920); Proc. Intern. Congr. App. Me&. (Delft), 55 (1924). 2. A. JOFFE, The Physics cf Crystals. McGraw-Hill, New York (1928). 3. R. A. LAD, J. A&. Phys. 93, 800 (1952). 4. F. I. METZ and R. A. LAD, J. Phys. Chem. 60, 277 (1956). 5. J. E. LENNARD-JONESand B. M. DENT, Proc. Roy. Sot. Lond. 121, 247 (1928). 6. J. J. GILMAN, J. Metals (Tmnmctions) 9, 449 (1957). 7. E. AERTS and W. DEKEYSER, Acta Met. 4, 557 (1956). 8. A. E. GORUM. E. R. PARKER and J. A. PASK. Paoer presented at American Ceramic Society Meeting ai Alfred University, Alfred, N.Y. (Sept. 9-10, 1957). 9. H. LEIDER, Phys. Rev. 101,56 (1956).

crystals

are stored after cleavage

in which the

has an importatit

bearing on the retention of ductility. Crystals of NaCl were found to remain ductile in vacuum or helium or if stored under oil.

They rapidly

become

brittle

if

exposed to gaseous nitrogen or oxygen. The

low

rate

temperature

of microcrack

formation

and the atmospheric

at room

effects on ductility

raise questions regarding the adequacy

of the original

postulate that the presence of microcracks

is the main

cause for the brittle behavior of ionic crystals. The data may be taken to indicate that ductility is restricted

by

additional

mechanisms.

with atmospheric

gases.

It is possible, however, that the limited body of information available at present could still be interpreted in terms of microcracks. The rate of cracking was measured in water polished crystals, whereas the relatively rapid reduction measured on freshly cleaved

Laboratory References

Received February 10, 1958.

One of these

might be, for example, the restriction of dislocation motion by the presence of surface layers formed by interaction

Lewis Flight Prop&on

in ductility has been crystals. Because of

Dilatational

strain due to dislocations in copper

Clarebrough, Hargreaves, and West(l) have recently measured both the stored energy and the macroscopic density changes in some specimens of plastically deformed copper. They have interpreted the density changes in terms of densities of dislocations by using a calculation of Stehle and Seeger.t2) The latter

ACTA

612

authors

have

calculated

the

with a screw dislocation

METALLURGICA,

dilatation

also used Cottrell’s(3) calculation from

their

of the elastic energy

energy

measurements.

densities The

dis-

location densities found in this way agree with those derived from the dilatational effects. The purpose of the

present

observed

communication

relationship

is to

between

energy can be obtained dynamic

arguments,

1958

associated

to estimate the dislocation

stored

6,

(5)

in copper on the basis of a

detailed model of the atomic interactions in the vicinity of the dislocation. Clarebrough et aZ.(l) have of a dislocation

VOL.

show

dilatation

approximately

without

that

the

where @ is the isothermal compressibility. We will apply (5) to th e present problem, interpreting E as the shear strain energy of a dislocation, and taking (a In ,u/&)~

from the work of Lazarus.c4)

TABLE 1. Summary of the results of Lazarus Elastic constant

Value (dyn/cm?

and stored

from thermo-

reference to an atomic

(~@h#/\

7.56 x 10” 2.36 x 10”

&(C,;%,,)

(3 In IJ/%)F (cms/cal)

n

0.11 x lo-” 0.23 x 10-l’

~

for copper. B

0.17 x 10-4 0.67 x lo-”

~

model of a dislocation. The results of Lazarus for copper are summarized

The elastic continuum model of a dislocation, which was used by Cottrell,(3) will also be used here.

in Table

In this model the elastic energy of a dislocation

dynamic arguments assume an isotropic solid, whereas

to a pure shear strain in the crystal. origin of the dilatational dislocation

The qualitative

strain associated

is then as follows:

is due

with the

It has been shown by

Lazarust4) that the elastic shear modulus of a crystal is changed by dilatation

of the latter.

energy of a crystal subjected

Therefore the

to an elastic shear strain

can be lowered by changing the volume of the crystal in such a way as to lower the shear modulus. In order to describe thermodynamic

this effect by a quantitative

relation, we consider a system which

was analyzed by Zenerc5) in another connection, namely, a cylindrical specimen of radius R and length L, subjected

to a torque

and to a hydrostatic entire

surface.

r applied at the two ends,

pressure

The

specimen

2, applied

d(E+pV-

TS)=

to

where E is the energy of the specimen,

V is its volume

and S its entropy, and 4 is the angle of twist. (I) is a perfect differential, the relation

Since

Following

Zener,

work done by the torque

we denote

in twisting

by

E the

the specimen

from equilibrium to 4 at constant pressure temperature, and divide both sides of (2) by r:

and

for r the formula of elastic theory i?pRi r=2L-

4

of copper, the exact value of (a In &n),

which

It seems likely

that most of the shear will be of the type

requires

the least energy,

namely,

that

which the shear modulus is lowest. Since $(C,, is less than one-third +(C,, -

C,,)

for C,,)

as great as C,,, the figures for

will be used in comparing equation (5)

with the experiments. TABLE 2. Comparison of the observed dilatation”’ with that calculated from equation (5), using the pressure dependence of &(C,, - Cl,). o/0Deformation

Stored energy (cal/cm5)

AVIV Calculated

AVIV Observed

0.57 x 10-a 0.76 x lo-” 0.88 x 10-e

0.91 x 10-4 1.38 x 1O-4 1.93 x IO-”

/ 30

0.85 1.14 1.31

::

~

~

The comparison is given in Table 2. It is seen that the calculated dilatations are too small by roughly a factor

two.

In view

of a dislocation

of the fact can only

that

any elastic

be approximate,

because much of the energy is stored in regions where the strain is far beyond the elastic limit, this agreement is as good as can be expected. Xemiconductor and Solid State

R. W. KEYES

Physics Department Westinghowe Research Laboratories Pittsburgh 35, Penn.

(3) Substituting

however,

theory

(2) is satisfied.

properties

be

(1)

model and our thermo-

to be used in (5) is not apparent.

for this system

Vdp+rdf$-#ST

Since Cottrell’s

two shear moduli are required to describe the elastic

over the

is considered

elastically isotropic. Zener derives the perfect differential form

1.

(4)

References 1. L. M. CLAREBROUOH, M. E. HARGREAVES and G. W. WEST, Acta Met. 5, 738 (1957). 2. H. STEELEand A. SEEGER,2. Phys. 148, 217 (1956). 3. A. H. COTTRELL,Dislocations and Plastic Flow in Crystals. Clarendon Press, Oxford (1953). 4. D. LAZARUS,Phys. Rev. 76, 545 (1949). 5. C. ZENER, Acta Cryst. 2, 163 (1949). Received February 14, 1958.